Right Triangle - Characterizations

Characterizations

A triangle ABC with sides, semiperimeter s, area T, altitude h opposite the longest side, circumradius R, inradius r, exradii r_a, r_b, r_c (tangent to a, b, c respectively), and medians m_a, m_b, m_c is a right triangle if and only if any one of the statements in the following six categories is true. All of them are of course also properties of a right triangle, since characterizations are equivalences.